Question

A transformer has 800 turns in the primary coil and 40 turns in the secondary coil. a) What happens if an \(\mathrm{AC}\) voltage of \(100 . \mathrm{V}\) is across the primary coil? b) If the initial AC current is \(5.00 \mathrm{~A}\), what is the output current? c) What happens if a DC current at \(100 .\) V flows into the primary coil? d) If the initial DC current is \(5.00 \mathrm{~A}\), what is the output current?

Short answer

What happens when a DC voltage and current are applied to the primary coil? Answer: When an AC voltage of 100V and AC current of 5.00 A are applied to the primary coil, the voltage at the secondary coil will be 5V, and the output current in the secondary coil will be 100 A. When a DC voltage and current are applied to the primary coil, no voltage will be induced and no output current will occur in the secondary coil, as transformers do not function with DC voltages and currents.

Step-by-step solution

Recall the Turn Ratio Formula for Transformer

The turn ratio of a transformer is given by the ratio of the number of primary coil turns to the secondary coil turns. It is denoted as \(N_{p}/N_{s}\). The turn ratio formula for the transformer is: \(N_{p}/N_{s} = V_{p}/V_{s} = I_{s}/I_{p}\) Where: \(N_{p}\) - Number of turns in the primary coil \(N_{s}\) - Number of turns in the secondary coil \(V_{p}\) - AC voltage across the primary coil \(V_{s}\) - AC voltage across the secondary coil \(I_{p}\) - AC current in the primary coil \(I_{s}\) - AC current in the secondary coil

Calculate the Turn Ratio

Given that \(N_{p}=800\) and \(N_{s}=40\). We can now compute the turn ratio as follows: Turn Ratio = \(N_{p}/N_{s} = 800/40 = 20\)

Apply the Turn Ratio to Calculate the Voltage and Current (a and b)

Since the turn ratio is 20, we can now answer parts (a) and (b) using the turn ratio, applied voltages, and currents. a) Given \(V_{p} = 100~V\), we can compute \(V_{s}\) using the turn ratio formula: \(V_{s} = V_{p}/\mathrm{Turn~Ratio} = 100/20 = 5~V\) b) Given \(I_{p} = 5.00~A\), we can compute \(I_{s}\) using the turn ratio formula: \(I_{s} = I_{p} \times \mathrm{Turn~Ratio} = 5.00 \times 20 = 100~A\)

Discuss the DC Scenario (c and d)

Transformers only work with AC voltages, not DC voltages. This means that when a DC voltage is applied: c) No voltage will be induced in the secondary coil. The transformer will not function. d) Similarly, when a DC current flows in the primary coil, no current will flow in the secondary coil. The transformer will not function. To summarize the answers: a) The voltage across the secondary coil will be \(5~V\). b) The output current in the secondary coil will be \(100~A\). c) No voltage will be induced in the secondary coil when a DC current is applied. d) No output current will occur if a DC current is applied to the primary coil.

Key concepts

Turn Ratio Formula

The turn ratio is fundamental to understanding how transformers operate. In essence, it represents the relationship between the coils of a transformer. Specifically, it is the ratio of the number of turns in the primary coil (\(N_{p}\)) to the number of turns in the secondary coil (\(N_{s}\)). This ratio significantly affects how the input signal is transformed.

When you see a problem involving a transformer with a known number of turns on both its primary and secondary coils, your first step should be to calculate this ratio. For the given exercise, the transformer's turn ratio was figured out by dividing the turn count of the primary coil by the secondary, which in numbers was calculated as \(800 / 40 = 20\). This implies that for every 20 turns in the primary coil, there is only one turn in the secondary coil.

Using this turn ratio, you can determine how the transformer will affect voltage and current levels. As the turn ratio increases, it means the primary coil has many more turns than the secondary, and consequently, you would expect a step-down in voltage in the secondary coil. This core concept of the turn ratio enables you to grasp the changes a transformer will induce on an AC signal, both in terms of volts and amps.

AC Voltage Transformation

When working with transformers in an AC circuit, voltage transformation is a crucial concept. The transformation of voltage from the primary to the secondary coil of a transformer is directly related to the turn ratio. In this particular exercise, with a turn ratio of 20:1 and an input or primary voltage (\(V_{p}\)) of 100 volts, the output or secondary voltage (\(V_{s}\)) is expected to be reduced by the same ratio.

Applying the formula \(V_{s} = V_{p} / \text{Turn Ratio}\) yields an output voltage of 5 volts. It's important to note that the voltage is transformed proportionally to the turn ratio, reflecting the transformer's ability to 'step down' the voltage. This principle applies only to AC circuits because transformers rely on the changing magnetic fields produced by alternating current. This concept is integral to the functionality of a wide array of electrical systems, from power distribution to the adapters you use for electronic devices.

Transformer Function with DC Current

Transformers are designed to work exclusively with alternating current (AC). The reason lies in how they operate: transformers rely on the principle of electromagnetic induction, which requires a varying magnetic field to induce voltage across the coils. With direct current (DC), the magnetic field is static, which means it doesn't change with time and, as a result, can't induce a voltage in the secondary coil.

In the context of the exercise, introducing DC current to the primary coil resulted in no action—no voltage is induced in the secondary coil, and therefore, no current flows through it. This highlights an important limitation of transformers: they cannot be used to 'transform' or modify DC voltage levels in the same way that they can with AC. This differentiation is crucial for understanding the proper applications of transformers and why they are so prevalent in AC power distribution systems but absent in DC circuits.